On the sampling Lovász Local Lemma for atomic constraint satisfaction problems

by   Vishesh Jain, et al.

We study the problem of sampling an approximately uniformly random satisfying assignment for atomic constraint satisfaction problems i.e. where each constraint is violated by only one assignment to its variables. Let p denote the maximum probability of violation of any constraint and let Δ denote the maximum degree of the line graph of the constraints. Our main result is a nearly-linear (in the number of variables) time algorithm for this problem, which is valid in a Lovász local lemma type regime that is considerably less restrictive compared to previous works. In particular, we provide sampling algorithms for the uniform distribution on: (1) q-colorings of k-uniform hypergraphs with Δ≲ q^(k-4)/3 + o_q(1). The exponent 1/3 improves the previously best-known 1/7 in the case q, Δ = O(1) [Jain, Pham, Vuong; arXiv, 2020] and 1/9 in the general case [Feng, He, Yin; STOC 2021]. (2) Satisfying assignments of Boolean k-CNF formulas with Δ≲ 2^k/5.741. The constant 5.741 in the exponent improves the previously best-known 7 in the case k = O(1) [Jain, Pham, Vuong; arXiv, 2020] and 13 in the general case [Feng, He, Yin; STOC 2021]. (3) Satisfying assignments of general atomic constraint satisfaction problems with p·Δ^7.043≲ 1. The constant 7.043 improves upon the previously best-known constant of 350 [Feng, He, Yin; STOC 2021]. At the heart of our analysis is a novel information-percolation type argument for showing the rapid mixing of the Glauber dynamics for a carefully constructed projection of the uniform distribution on satisfying assignments. Notably, there is no natural partial order on the space, and we believe that the techniques developed for the analysis may be of independent interest.



There are no comments yet.


page 1

page 2

page 3

page 4


Towards the sampling Lovász Local Lemma

Let Φ = (V, 𝒞) be a constraint satisfaction problem on variables v_1,…, ...

Complexity of Reconfiguration Problems for Constraint Satisfaction

Constraint satisfaction problem (CSP) is a well-studied combinatorial se...

Finding Small Satisfying Assignments Faster Than Brute Force: A Fine-grained Perspective into Boolean Constraint Satisfaction

To study the question under which circumstances small solutions can be f...

Approximability of all Boolean CSPs in the dynamic streaming setting

A Boolean constraint satisfaction problem (CSP), Max-CSP(f), is a maximi...

Sampling Constraint Satisfaction Solutions in the Local Lemma Regime

We give a Markov chain based algorithm for sampling almost uniform solut...

Learning from Satisfying Assignments Using Risk Minimization

In this paper we consider the problem of Learning from Satisfying Assign...

Decentralized Constraint Satisfaction

We show that several important resource allocation problems in wireless ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.